Understand the DDPG algorithm in Keras
The answer to your first question is because the line 'update the critic by minimising the loss $L = \frac{1}{N} \sum_i \left( y_i - Q(s_i, a_i |\theta^Q)\right)^2$ is implying that you will do this by using a gradient, i.e. you calculate the gradient of the loss wrt the parameters and perform a gradient descent step.
For the second question, I am not 100% sure because I don't use tensorflow but I would assume that in the automatic differentiation used it will do the multiplication automatically behind the scenes. In PyTorch I know that it automatically calculates the chain rule when you take the derivative, so it is likely to be a reason similar to this.